#8001027

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10,14,15)(11,16,19,23)(12,17,20,24)(13,21,22,18)(25,26,31,33)(27,36,37,32)(28,29,34,39)(30,40,38,35);(1,2,5,6)(3,7,4,8)(10,15)(12,20)(16,23)(18,21)(25,27,31,37)(26,32,33,36)(28,35,34,40)(29,38,39,30);(1,3,5,4)(2,7,6,8)(10,15)(11,19)(17,24)(18,21)(26,33)(27,37)(28,34)(30,38);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,17)(14,19)(15,23)(20,21)(22,24)(25,28)(26,29)(27,35)(30,36)(31,34)(32,38)(33,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,18)(13,16)(14,20)(15,24)(19,21)(22,23)(25,29)(26,34)(27,30)(28,33)(31,39)(32,35)(36,40)(37,38);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,17)(12,16)(14,22)(15,21)(19,24)(20,23)(25,30)(26,35)(27,29)(28,36)(31,38)(32,34)(33,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908223);